Here \(N\)  is a globally timelike vector field such that \(N=T\)  away from the event horizon, \(d\mu _{\Sigma _{\tau }}\)  is the volume form corresponding to the hypersurface \(\Sigma _{\tau }\) , and \(n_{\tau }\)  the normal to \(\Sigma _{\tau }\)  (note that we will also use \(n\)  to denote the normal to other hypersurfaces in analogous situations without specifying it). First of all, the Dafermos–Rodnianski hierachy [1]} schematically reads as:
\(\int _{\tau }\int _{\Sigma _{\tau }}\frac{r^{p-1}}{r^2}\cdot J^{N}\left[\phi _{\ge \ell } \right] \cdot n_{\tau } \, d\mu _{\Sigma _{\tau }} d\tau \lesssim \int _{\Sigma _{\tau _0}}\frac{r^{p}}{r^2}\cdot J^{N}\left[\phi _{\ge \ell } \right] \cdot n_{\tau _0} \, d\mu _{\Sigma _{\tau _0}}\) 
